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A055022 Number of 1-punctured staircase polygons (by perimeter) with a hole of perimeter 4. 1
0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 92, 576, 3214, 16664, 82160, 390656, 1807781, 8192524, 36519556, 160645504, 699030226, 3014470024, 12901501696, 54863119744, 232022899306, 976598630968, 4093581923320, 17096805375360, 71176501409756 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,10

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..500

A. J. Guttmann et al., Punctured polygons and polyominoes on the square lattice, arXiv:cond-mat/0003441 [cond-mat.stat-mech], 2000.

A. J. Guttmann et al., Punctured polygons and polyominoes on the square lattice, J. Physics A: Math. and Gen, 33 (9) (2000), 1735-1764.

FORMULA

Conjecture: n*(n-8)*a(n) +2*(-4*n^2+35*n-45)*a(n-1) +8*(2*n-9)*(n-5)*a(n-2)=0. - R. J. Mathar, Aug 14 2012

For n>3, a(n) = 4^(n-4)-binomial(2n,n)(n-3)(n^2-5n+10)/(4(2n-1)(2n-3)(2n-5)). - Michael D. Weiner, Jan 17 2018

MAPLE

gf := (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)/(2*(1 - 4*x)) - (1 - 6*x + 10*x^2 - 4*x^3)/(2*sqrt(1 - 4*x)): s := series(gf, x, 50): for i from 0 to 50 do printf(`%d, `, coeff(s, x, i)) od:

MATHEMATICA

Join[{0, 0, 0, 0}, Table[4^(n - 4) - Binomial[2 n, n] (n - 3) (n^2 - 5 n + 10) / (4 (2 n - 1) (2 n - 3) (2 n - 5)), {n, 4, 50}]] (* Vincenzo Librandi, Jan 20 2018 *)

PROG

(MAGMA) [0, 0, 0, 0] cat [4^(n-4)-Binomial(2*n, n)*(n-3)*(n^2-5*n+10) div (4*(2*n-1)*(2*n-3)*(2*n-5)): n in [4..30]]; // Vincenzo Librandi, Jan 20 2018

CROSSREFS

Sequence in context: A246585 A120990 A220330 * A220937 A221291 A044263

Adjacent sequences:  A055019 A055020 A055021 * A055023 A055024 A055025

KEYWORD

easy,nonn

AUTHOR

James A. Sellers, May 31 2000

STATUS

approved

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Last modified May 26 05:25 EDT 2019. Contains 323579 sequences. (Running on oeis4.)